3.3.1 The Expected Value of X

expected value or mean value

• Let $X$ be a discrete random variable (rv) with:
  • Set of possible values $D$
  • Probability mass function (pmf) $p(x)$
• The expected value (or mean value) of $X$
  • denoted by $E(X)$, ${\mu }_X$, or simply $\mu$,
  • is defined as: $E(X)={\mu }_X={\sum }_{x\in D}x\cdot p(x)$

• This means that the expected value is the sum of each possible value $x$ multiplied by its corresponding probability $p(x)$.
• $E(X)$ represents the “average” or “center” of the distribution of $X$.

EX 3.16 continued table (3.6)

• In Example 3.16, the expected value $\mu =4.57$:

  • This value is not a possible value of $X$ (number of courses)

• Interpretation of “expected” value:

  • The term “expected” does not mean that we expect to see $X=4.57$ for a single randomly selected student
  • Since $X$ represents discrete values (number of courses), a student can only register for 1, 2, 3, …, or 7 courses, not 4.57

• The expected value $\mu$ represents the average or mean value over the entire population

  • It’s a statistical average, not a possible individual outcome

• The concept of expected value should be interpreted with caution, especially in cases where the expected value is not a feasible outcome for individual samples.

• EX 3.17 Apgar scale

• EX 3.18 expected value of Bernoulli random variable

• EX 3.19 first born boy

Example

• Interpretation of $\mu$:
  • The mean $\mu$ can be understood as the long-run average of observed values of a random variable $X$.
• Observational process:
  • Consider a sequence of observations:
    • First value: $x_1$
    • Second value: $x_2$
    • Third value: $x_3$
    • And so on.
  • After performing this process a large number of times, compute the sample average of the observed values: $$\text{Sample Average}=\frac{x_1+x_2+x_3+\dots +x_n}{n}$$
• Relationship to $\mu$:
  • This sample average will typically be close to $\mu$ as $n$ becomes large.
  • Thus, $\mu$ represents the expected long-term average outcome when the experiment is performed repeatedly.
• Example:
  • In the context of the Apgar scores, the long-run average score is given by: $$\mu =7.15$$
  • This means that, on average, the Apgar scores of a large number of newborns will converge to $7.15$.
• Conclusion:
  • Understanding $\mu$ as the long-run average provides insight into the expected behavior of the random variable $X$ across many trials.

EX 3.20 interview before job